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Haas theorem revisited | | Benoît Bertrand
; Erwan Brugallé
; Arthur Renaudineau
; | | Date: |
7 Sep 2016 | | Abstract: | Haas theorem describes all partchworkings of a given non-singular plane
tropical curve $C$ giving rise to a maximal real algebraic curve. The space of
such patchworkings is naturally a linear subspace $W_C$ of the
$mathbb{Z}/2mathbb{Z}$-vector space $overrightarrow Pi_C$ generated by the
bounded edges of $C$, and whose origin is the Harnack patchworking. The aim of
this note is to provide an interpretation of affine subspaces of
$overrightarrow Pi_C $ parallel to $W_C$. To this purpose, we work in the
setting of abstract graphs rather than plane tropical curves. We introduce a
topological surface $S_Gamma$ above a trivalent graph $Gamma$, and consider a
suitable affine space $Pi_Gamma$ of real structures on $S_Gamma$ compatible
with $Gamma$. We characterise $W_Gamma$ as the vector subspace of
$overrightarrow Pi_Gamma$ whose associated involutions induce the same
action on $H_1(S_Gamma,mathbb{Z}/2mathbb{Z})$. We then deduce from this
statement another proof of Haas’s original result. | | Source: | arXiv, 1609.1979 | | Services: | Forum | Review | PDF | Favorites | |
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